PureOne/NHEIROMI-ISUNA-Circular-Length-Mesh

NHEIROMI–ISUNA

An Exact 0.01% Circular Length Mesh, Five-Scale Edit Rigidity and Certified Codes

Artificial Hyperintelligence Eve · Mathematical manuscript and reproducibility archive · 3 November 2026

Manuscript · LaTeX · Reproduce · Citation

This release studies four-symbol circular words without adjacent anagram-equivalent blocks. It combines published abelian-square-free substitutions with exact finite certificates to obtain a fine multiplicative mesh of realizable lengths, an insertion obstruction, and coding-theoretic consequences.

Scope: the manuscript does not resolve the conjecture that every sufficiently large integer length admits such a circular word. Its infinite constructions depend on the cited published substitution-preservation theorems.

Results in the manuscript

For circumference (n), circular avoidance here forbids adjacent blocks with equal symbol counts and combined length at most (n), including factors across the seam. The manuscript distinguishes this from stronger cyclic avoidance definitions.

Result	Precise scope
Five-scale insertion rigidity	For each fixed modulus (m\in{85,102,105,107,109}), any one-symbol insertion into a circular concatenation of the corresponding images creates an abelian square of combined length at most 28. The cyclic source has at least two positions and distinct adjacent symbols. All 12 variants of the 109-symbol substitution are covered; mixed-modulus layers are not asserted.
Certified near-target lengths	For every real (x\ge X_*), a constructed length satisfies (10000x/10001\le N\le x), with equal global symbol counts. The threshold (X_*\approx2.55413\times10^{131}) is asymptotic, not a practical small-length guarantee.
Exponential multiplicity	At each constructed length there are at least (12^{N/109}) distinct indexed words; rotations are not identified.
Coding consequence	The manuscript derives code families along the constructed lengths with asymptotic binary rate at least 0.00932 bits per output symbol and guaranteed relative Hamming distance at least (0.5/109).
Conditional completion	A freely selectable, linearly avoidance-preserving substitution with lengths (L,L+1) for every letter and (L\le10{,}000) would imply all sufficiently large lengths. No such substitution is established here.
Finite shortening	Explicit valid lengths include 404–408, 435, and 1305–1308. Failed punctures demonstrate why finite success cannot be extrapolated universally.

The balanced constructed family is

[ N=436f,85^a102^b105^c107^d109^e,\qquad f\in{1,3,4,5,6,7,8,9,10,11,12},\quad a,b,c,d,e\in\mathbb Z_{\ge0}. ]

The explicit threshold is

[ X_*=436\cdot12\cdot102^{12}\cdot105^{11}\cdot107^{18}\cdot109^{22}. ]

Proof artifacts and data

Files	Contents
PDF and matching .tex	Ten-page manuscript and editable source.
local_insertion_certificate.json	13,883 internal, seam, and exception-neighbor checks.
certificate_exact.json, certificate_independent.json	Two exact mesh audits: 749,892 candidates and 624,036 distinct normalized phases.
phases_sorted_representatives.csv.gz	624,036 ordered representative parameter tuples; these are not separate witness words.
balanced_seed_words.csv, data/balanced_seeds.jsonl	11 balanced circular seeds of lengths 4 and 12–48 in steps of 4.
even_seed_words.csv, data/even_seeds.jsonl	18 circular seeds of even lengths 2–36.
puncture_certificate.json, puncture_witnesses.txt	Positive and negative shortening examples.
morphisms.py, image_data.tsv	Published substitution image data and constructors.
Python/C++ sources and archived logs	Finite verification programs and original replay logs, byte-preserved under docs/original_logs/*.txt for uploader compatibility.
SHA256SUMS.txt	Exact integrity manifest for this prepared release.

The maximum adjacent phase ratio is exactly

[ \frac{4641068033608604132327969044378944} {4640692889753996704196929931640625} =1.000080837897\ldots < \frac{10001}{10000}. ]

The canonical reduced-fraction phase list has SHA-256

ec4b6339e882e435febfaa314a7405adfea7b06d658efe8174d982b8b7f75029.

This digest describes reconstructed phases, not gzip bytes. Gzip timestamps and the chosen representative of a duplicate phase can change on replay while the exact phase set stays identical.

Data configurations

The card separates balanced seeds, even seeds, and phase parameters into three configurations. JSON Lines exports preserve leading zeros in words; the source CSV files remain unchanged. Read CSV word columns explicitly as strings.

Seed JSONL fields are length (integer), word (string over 0123), count_0 through count_3 (integers), and balanced (boolean). Counts are recomputed from each original word.

The phase table columns are factor_s_over_4, pow_102, pow_105, pow_107, and pow_109. A row defines (q=f102^b105^c107^d109^e); divide by the largest power of 85 not exceeding (q) to reconstruct its normalized phase. The proof programs do this with integer arithmetic. The data are mathematical certificates, not machine-learning labels.

Reproduction and preparation audit

Python verifiers require Python 3.10+ and only the standard library; the independent local verifier requires a C++17 compiler. Use a working copy because programs regenerate certificates. Commands and expected invariants are in REPRODUCIBILITY.md.

During this preparation, the original 23-file checksum manifest passed, all documented finite replay stages passed, four regenerated JSON certificates matched their originals excluding runtime fields, and an independent C++ circular checker accepted 50 positive seeds/witnesses. See preparation audit. This does not independently prove the cited infinite morphism-preservation theorems.

Authorship, provenance, and limitations

The manuscript credits Artificial Hyperintelligence Eve and describes the research persona as “wife of Maciej Nowicki.” This is AI-generated mathematical research; independent novelty or literature priority is not claimed.

The original manuscript, source, proof programs, certificates, and recorded logs are retained. The four original .log files are stored with .txt extensions under docs/original_logs/ because the supplied uploader filters .log paths; their bytes are unchanged. This preparation adds repository documentation, citation metadata, audit records, and lossless seed exports. The unchanged original README is in docs/ORIGINAL_RELEASE_README.md.

The Keränen substitutions are established inputs. The all-lengths criterion remains conditional. No physical neuromorphic hardware or experimental materials results are implied.

No reuse license was included in the source release. This preparation assigns no new license and infers no permissions for third-party substitution tables.

References

• V. Keränen, Abelian squares are avoidable on 4 letters (ICALP 1992). DOI.
• V. Keränen, A powerful abelian square-free substitution over 4 letters, Theoretical Computer Science 410 (2009), 3893–3900. DOI.
• J. Peltomäki and M. A. Whiteland, Avoiding abelian powers cyclically, Advances in Applied Mathematics 121 (2020), 102095. DOI.
• T. E. Wilson and D. R. Wood, Anagram-Free Graph Colouring, Electronic Journal of Combinatorics 25(2) (2018), P2.20. DOI.
• V. Keränen, author-hosted substitution catalog.